Lemma 28.20.2 (05P2)—The Stacks project

Lemma 28.20.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent:

$\mathcal{F}$ is a flat $\mathcal{O}_ X$-module of finite presentation,
$\mathcal{F}$ is $\mathcal{O}_ X$-module of finite presentation and for all $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module,
$\mathcal{F}$ is a locally free, finite type $\mathcal{O}_ X$-module,
$\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module, and
$\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type, for every $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module, and the function

\[ \rho _\mathcal {F} : X \to \mathbf{Z}, \quad x \longmapsto \dim _{\kappa (x)} \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]

is locally constant in the Zariski topology on $X$.

Proof. This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 10.78.2. The translation uses Lemmas 28.16.1, 28.16.2, 28.19.1, and 28.20.1. $\square$

Comments (2)

Comment #4607 by James Waldron on October 18, 2019 at 06:23

Unless this appears elsewhere, it might be useful to add that if $X$ is reduced then one can weaken statement (5) to ' $\mathcal{F}$ is finite type and $\rho_{\mathcal{F}}$ is locally constant'. i.e. the requirement that $\mathcal{F}_x$ is free can be dropped.

(This appears as Exercise 13.7.K in the 17/11/2017 version of Vakil's notes The Rising Sea or Exercise 5.8 in Hartshorne.)

The affine case Lemma 00NX could also include the corresponding statement for finite type modules over reduced rings.

Comment #4777 by Johan on December 09, 2019 at 15:25

Thanks! I have added this as a separate lemma in both the algebra and schemes case. See changes.

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